Our new idea here is the proposal of a linear clustering process (LCP) on a graph, where nodes move in one-dimensional space and tend to concentrate in groups that lead to network communities. Linear means "proportional to the graph", which is needed, because the aim is to cluster the graph and the process should only help and not distract from our main aim of clustering. A non-linear process depends intricately on the underlying graph that we want to cluster and may result in worse clustering! Our LCP leads to a new and non-trivial graph matrix \(W\)  in (REF ) in Theorem REF , whose spectral decomposition is at least as good as the best clustering result, based on the non-back tracking matrix [1]}. Moreover, the new graph matrix \(W\)  has a more "natural" relation to clustering than the non-back tracking matrix, that was not designed for clustering initially. Finally, our resulting LCP clustering algorithm seems surprisingly effective and can compete computationally with any other clustering algorithm, while achieving generally a better result!
